5-manifolds: 1-connected
Line 1: | Line 1: | ||
+ | == Introduction == | ||
<wikitex>; | <wikitex>; | ||
− | Let $\mathcal{M}_{5}(e)$ be the set of diffeomorphism classes of [[wikipedia:Closed_manifold|closed]], [[wikipedia:Oriented_manifold#Orientability_of_manifolds|oriented]], [[wikipedia:Differentiable_manifold|smooth]], [[wikipedia:Simply-connected|simply-connected]] [[wikipedia:5-manifold|5-manifolds]] $M$ and let $\mathcal{M}_5^{\Spin}(e)\subset \mathcal{M}_5(e)$ be the subset of diffeomorphism classes of [[wikipedia:Spin_manifold|spinable manifolds]]. | + | Let $\mathcal{M}_{5}(e)$ be the set of diffeomorphism classes of [[wikipedia:Closed_manifold|closed]], [[wikipedia:Oriented_manifold#Orientability_of_manifolds|oriented]], [[wikipedia:Differentiable_manifold|smooth]], [[wikipedia:Simply-connected|simply-connected]] [[wikipedia:5-manifold|5-manifolds]] $M$ and let $\mathcal{M}_5^{\Spin}(e)\subset \mathcal{M}_5(e)$ be the subset of diffeomorphism classes of [[wikipedia:Spin_manifold|spinable manifolds]]. The calculation of $\mathcal{M}_5^{\Spin}(e)$ was first obtained by Smale in {{cite|Smale1962}} and was one of the first applications of the [[wikipedia:H-cobordism|h-cobordism theorem]]. A little latter Barden, {{cite|Barden1965}}, applied a clever surgery argument and results of {{cite|Wall1964}} on the diffeomorphism groups of $4$-manifolds to give an explicit and complete classification of all of $\mathcal{M}_{5}(e)$. |
+ | |||
+ | Simply-connected $5$-manifolds are an appealing class of manifolds: the dimension is just large enough so that the full power of surgery techniques can be applied but it is low enough that the manifolds are simple enough to be readily classified. A feature of simply-connected $5$-manifolds is that the homotopy, homeomorphism and diffeomorphism classification all coincide. Note that every simply-connected $5$-dimensional [[wikipedia:Poincaré_space|Poincaré space]] is smoothable. The classification of of simply-connected Poincaré spaces may be found in {{cite|Stöcker1982}}. | ||
</wikitex> | </wikitex> | ||
Line 6: | Line 9: | ||
<wikitex>; | <wikitex>; | ||
We first list some familiar 5-manifolds using Barden's notation: | We first list some familiar 5-manifolds using Barden's notation: | ||
− | * $X_0 \coloneq S^5$ | + | * $X_0 \coloneq S^5$. |
− | * $M_\infty \coloneq S^2 \times S^3$ | + | * $M_\infty \coloneq S^2 \times S^3$. |
− | * $X_\infty \coloneq S^2 \times_{\gamma} S^3$, the total space of the non-trivial $S^3$-bundle over $S^2$ | + | * $X_\infty \coloneq S^2 \times_{\gamma} S^3$, the total space of the non-trivial $S^3$-bundle over $S^2$. |
* $X_{-1} \coloneq \SU_3/\SO_3$, the Wu-manifold, is the homogeneous space obtained from the standadard inclusion of $\SO_3 \rightarrow SU_3$. | * $X_{-1} \coloneq \SU_3/\SO_3$, the Wu-manifold, is the homogeneous space obtained from the standadard inclusion of $\SO_3 \rightarrow SU_3$. | ||
− | * Next we present a construction of Spin 5-manifolds. Note that all homology groups are with integer coefficients. Given a finitely generated abelian group $G$, let $X_G$ denote the degree 2 Moore space with $H_2(X_G) = G$. The space $X_G$ may be realised as a finite CW-complex and so there is an embedding $X_G\to\Rr^6$. Let $N(G)$ be a [[regularneighbourhood|regular neighbourhood]] of $X_G\subset\Rr^6$ and let $M_G$ be the boundary of $N(G)$. Then $M_G$ is a closed, smooth, simply-connected, spinable 5-manifold with $H_2(M_G)\cong G \oplus TG$ where $TG$ is the torsion subgroup of $G$. For example, $M_{\Zz^r} \cong \sharp_r S^2 \times S^3$ where $\sharp_r$ denotes the $r$-fold connected sum. | + | * Next we present a construction of Spin 5-manifolds. Note that all homology groups are with integer coefficients. Given a finitely generated abelian group $G$, let $X_G$ denote the degree 2 Moore space with $H_2(X_G) = G$. The space $X_G$ may be realised as a finite CW-complex and so there is an embedding $X_G\to\Rr^6$. Let $N(G)$ be a [[regularneighbourhood|regular neighbourhood]] of $X_G\subset\Rr^6$ and let $M_G$ be the boundary of $N(G)$. Then $M_G$ is a closed, smooth, simply-connected, spinable 5-manifold with $H_2(M_G)\cong G \oplus TG$ where $TG$ is the torsion subgroup of $G$. For example, $M_{\Zz^r} \cong \sharp_r S^2 \times S^3$ where $\sharp_r$ denotes the $r$-fold connected sum. |
* For the non-Spin case let $(G, w)$ be a pair with $w\co G \to\Zz_2$ a surjective homomorphism and $G$ as above. We shall construct a non-Spin 5-manifold $M_{(G, w)}$ with $H_2(M_{(G, w)}) \cong G \oplus TG$ and [[#Invariants|second Stiefel-Whitney class]] $w_2$ given by $w$ composed with the projection $G \oplus TG \to G$. If $(G, w) = (\Zz, 1)$ let $N_{(\Zz, 1)}$ be the non-trivial $D^4$-bundle over $S^2$ with boundary $\partial N_{(\Zz, 1)} = M_{(\Zz, 1)} = X_{\infty}$. If $(G, w) = (\Zz, 1) \oplus (\Zz^r, 0)$ let $N_{(G, w)}$ be the boundary connected sum $N_{(\Zz, 1)} \natural_r S^2 \times D^4$ with boundary $M_{(G, w)} = X_{\infty} \sharp_r S^2 \times S^3$. In the general case, present $G = F/i(R)$ where $i \co R \to F$ is an injective homomorphism between free abelian groups. Lift $(G, w)$ to $(F, \bar w)$ and observe that there is a canonical identification $F = H_2(M_{(F, \bar w)})$. If $\{r_1, \dots, r_n \}$ is a basis for $R$ note that each $i(r_i) \in H_2(M_{(F, \bar w)})$ is represented by a an embedded 2-sphere with trivial normal bundle. Let $N_{(G, w)}$ be the manifold obtained by attaching 3-handles to $N_{(F, \bar w)}$ along spheres representing $i(r_i)$ and let $M_{(G, w)} = \partial N_{(G, w)}$. One may check that $M_{(G, w)}$ is a non-Spin manifold as described above. | * For the non-Spin case let $(G, w)$ be a pair with $w\co G \to\Zz_2$ a surjective homomorphism and $G$ as above. We shall construct a non-Spin 5-manifold $M_{(G, w)}$ with $H_2(M_{(G, w)}) \cong G \oplus TG$ and [[#Invariants|second Stiefel-Whitney class]] $w_2$ given by $w$ composed with the projection $G \oplus TG \to G$. If $(G, w) = (\Zz, 1)$ let $N_{(\Zz, 1)}$ be the non-trivial $D^4$-bundle over $S^2$ with boundary $\partial N_{(\Zz, 1)} = M_{(\Zz, 1)} = X_{\infty}$. If $(G, w) = (\Zz, 1) \oplus (\Zz^r, 0)$ let $N_{(G, w)}$ be the boundary connected sum $N_{(\Zz, 1)} \natural_r S^2 \times D^4$ with boundary $M_{(G, w)} = X_{\infty} \sharp_r S^2 \times S^3$. In the general case, present $G = F/i(R)$ where $i \co R \to F$ is an injective homomorphism between free abelian groups. Lift $(G, w)$ to $(F, \bar w)$ and observe that there is a canonical identification $F = H_2(M_{(F, \bar w)})$. If $\{r_1, \dots, r_n \}$ is a basis for $R$ note that each $i(r_i) \in H_2(M_{(F, \bar w)})$ is represented by a an embedded 2-sphere with trivial normal bundle. Let $N_{(G, w)}$ be the manifold obtained by attaching 3-handles to $N_{(F, \bar w)}$ along spheres representing $i(r_i)$ and let $M_{(G, w)} = \partial N_{(G, w)}$. One may check that $M_{(G, w)}$ is a non-Spin manifold as described above. | ||
− | * | + | * In {{cite|Barden1965|Section 1}} a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles $M \cong W \cup_f W$ where $W$ is a certain simply connected $5$-manifold with boundary $\partial W$ a simply-connected $4$-manifold and $f: \partial W \cong \partial W$ is a diffeomorphism. Barden used results of {{cite|Wall1964}} to show that diffeomorphisms realising the required ismorphisms of $H_2(\partial W)$ exist. |
− | * !!! To do: determine which 1-connected 5-manifolds appear as Brieskorn varieties. | + | |
+ | <!--* !!! To do: determine which 1-connected 5-manifolds appear as Brieskorn varieties. --> | ||
</wikitex> | </wikitex> | ||
Line 28: | Line 32: | ||
* $b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz$, the [[wikipedia:Poincaré duality#Bilinear_pairings_formulation|linking form]] of $M$ which is a non-singular anti-symmetric bi-linear pairing on $TH_2(M)$. | * $b_M \co TH_2(M) \times TH_2(M) \rightarrow \Qq/\Zz$, the [[wikipedia:Poincaré duality#Bilinear_pairings_formulation|linking form]] of $M$ which is a non-singular anti-symmetric bi-linear pairing on $TH_2(M)$. | ||
− | By {{cite|Wall1962|Proposition 1 & 2}} the linking form satisfies the identity $b_M(x, x) = w_2(x)$ where we regard $w_2(x)$ as an element of $\{0, 1/2\} \subset \Qq/\Zz$. | + | By {{cite|Wall1962|Proposition 1 & 2}} the linking form satisfies the identity $b_M(x, x) = w_2(x)$ where we regard $w_2(x)$ as an element of $\{0, 1/2\} \subset \Qq/\Zz$. |
For example, the Wu-manifold $X_{-1}$ has $H_2(X_{-1}) = \Zz_2$, non-trivial $w_2$ and $h(X_{-1}) = 1$. | For example, the Wu-manifold $X_{-1}$ has $H_2(X_{-1}) = \Zz_2$, non-trivial $w_2$ and $h(X_{-1}) = 1$. | ||
Line 34: | Line 38: | ||
An abstract non-singular, anti-symmetric linking form $b \co H \times H \rightarrow \Qq/\Zz$ on a finite group $H$ is a bi-linear function such that $b(x, y) = -b(y, x)$ and $b(x, y) = 0$ for all $y \in H$ if and only if $x = 0$. By {{cite|Wall1963}} such linking forms are classified up to isomorphism by the homomorphism $H \rightarrow \Zz_2, x \mapsto b(x, x)$. Moreover $H$ must be isomorphic to $T \oplus T$ or $T \oplus T \oplus \Zz_2$ for some finite group $T$ with $b(x,x) = 1/2$ if $x$ generates the $\Zz_2$ summand. In particular the second Stiefel-Whitney class of a 5-manifold $M$ determines the isomorphism class of the linking form $b_M$ and we see that the torsion subgroup of $H_2(M)$ is of the form $TH_2(M) \cong T \oplus T$ if $h(M) \neq 1$ or $TH_2(M) \cong T \oplus T \oplus \Zz_2$ if $h(M) = 1$ in which case the $\Zz_2$ summand is an orthogonal summand of $b_M$. | An abstract non-singular, anti-symmetric linking form $b \co H \times H \rightarrow \Qq/\Zz$ on a finite group $H$ is a bi-linear function such that $b(x, y) = -b(y, x)$ and $b(x, y) = 0$ for all $y \in H$ if and only if $x = 0$. By {{cite|Wall1963}} such linking forms are classified up to isomorphism by the homomorphism $H \rightarrow \Zz_2, x \mapsto b(x, x)$. Moreover $H$ must be isomorphic to $T \oplus T$ or $T \oplus T \oplus \Zz_2$ for some finite group $T$ with $b(x,x) = 1/2$ if $x$ generates the $\Zz_2$ summand. In particular the second Stiefel-Whitney class of a 5-manifold $M$ determines the isomorphism class of the linking form $b_M$ and we see that the torsion subgroup of $H_2(M)$ is of the form $TH_2(M) \cong T \oplus T$ if $h(M) \neq 1$ or $TH_2(M) \cong T \oplus T \oplus \Zz_2$ if $h(M) = 1$ in which case the $\Zz_2$ summand is an orthogonal summand of $b_M$. | ||
</wikitex> | </wikitex> | ||
− | + | ||
== Classification == | == Classification == | ||
<wikitex>; | <wikitex>; | ||
We first present the most economical classifications of $\mathcal{M}^{\Spin}_5(e)$ and $\mathcal{M}_5(e)$. Let ${\mathcal Ab}^{T \oplus T \oplus *}$ be the set of isomorphism classes finitely generated abelian groups $G$ with torsion subgroup $TG \cong H \oplus H \oplus C$ where $C$ is trivial or $C \cong \Zz_2$ and write ${\mathcal Ab}^{T \oplus T}$ and ${\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ for the obvious subsets. | We first present the most economical classifications of $\mathcal{M}^{\Spin}_5(e)$ and $\mathcal{M}_5(e)$. Let ${\mathcal Ab}^{T \oplus T \oplus *}$ be the set of isomorphism classes finitely generated abelian groups $G$ with torsion subgroup $TG \cong H \oplus H \oplus C$ where $C$ is trivial or $C \cong \Zz_2$ and write ${\mathcal Ab}^{T \oplus T}$ and ${\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ for the obvious subsets. | ||
− | {{beginthm|Theorem|{{cite|Smale1962|}}}} There is a bijective correspondence $$\mathcal{M}_5^{\Spin}(e) \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].$$ | + | {{beginthm|Theorem|{{cite|Smale1962|}}}} There is a bijective correspondence $$\mathcal{M}_5^{\Spin}(e) \rightarrow {\mathcal Ab}^{T\oplus T}, \quad [M] \mapsto [H_2(M)].$$ |
{{endthm}} | {{endthm}} | ||
− | {{beginthm|Theorem|{{cite|Barden1965}}}} The mapping | + | {{beginthm|Theorem|{{cite|Barden1965}}}} The mapping |
$$\mathcal{M}_{5}(e) \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) | $$\mathcal{M}_{5}(e) \rightarrow {\mathcal Ab}^{T \oplus T \oplus *} \times (\Nn \cup \{ \infty \}) | ||
, \quad [M] \mapsto ([H_2(M)], h(M))$$ | , \quad [M] \mapsto ([H_2(M)], h(M))$$ | ||
− | is an injection onto the subset of pairs $([G], n)$ where $[G] \in {\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ if and only if $n = 1$. | + | is an injection onto the subset of pairs $([G], n)$ where $[G] \in {\mathcal Ab}^{T \oplus T \oplus \Zz_2}$ if and only if $n = 1$. |
− | {{endthm}} | + | {{endthm}} |
The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms. | The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms. | ||
Line 68: | Line 72: | ||
* $X_0 \coloneq S^5$, $M_\infty\coloneq S^2 \times S^3$, $X_\infty\coloneq S^2 \times_{\gamma} S^3$, $X_{-1} \coloneq \SU_3/\SO_3$. | * $X_0 \coloneq S^5$, $M_\infty\coloneq S^2 \times S^3$, $X_\infty\coloneq S^2 \times_{\gamma} S^3$, $X_{-1} \coloneq \SU_3/\SO_3$. | ||
* For $1 < k < \infty$, $M_k = M_{\Zz_k}$ is the Spin manifold with $H_2(M) = \Zz_k \oplus \Zz_k$ constructed [[#Examples_and_constructions|above]]. | * For $1 < k < \infty$, $M_k = M_{\Zz_k}$ is the Spin manifold with $H_2(M) = \Zz_k \oplus \Zz_k$ constructed [[#Examples_and_constructions|above]]. | ||
− | * For $1 < j < \infty$ let $X_j = M_{(\Zz_{2^j}, 1)}$ constructed [[#Examples_and_constructions|above]] be the non-Spin manifold with $H_2(X_j) \cong \Zz_{2^j} \oplus \Zz_{2^j}$. | + | * For $1 < j < \infty$ let $X_j = M_{(\Zz_{2^j}, 1)}$ constructed [[#Examples_and_constructions|above]] be the non-Spin manifold with $H_2(X_j) \cong \Zz_{2^j} \oplus \Zz_{2^j}$. |
− | With this notation {{cite|Barden1965|Theorem 2.3}} states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by | + | With this notation {{cite|Barden1965|Theorem 2.3}} states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by |
$$ X_{j, k_1, \dots , k_n} = X_j \sharp M_{k_1} \sharp \dots \sharp M_{k_n}$$ | $$ X_{j, k_1, \dots , k_n} = X_j \sharp M_{k_1} \sharp \dots \sharp M_{k_n}$$ | ||
where $-1 \leq j \leq \infty$, $1 < k_i$, $k_i$ divides $k_{i+1}$ or $k_{i+1} = \infty$ and $\sharp$ denotes the [[wikipedia:Connected_sum|connected sum]] of oriented manifolds. The manifold $X_{j', k_1', \dots k_n'}$ is diffeomorphic to $X_{j, k_1, \dots, k_n}$ if and only if $(j', k'_1, \dots, k'_n) = (j, k_1, \dots, k_n)$. | where $-1 \leq j \leq \infty$, $1 < k_i$, $k_i$ divides $k_{i+1}$ or $k_{i+1} = \infty$ and $\sharp$ denotes the [[wikipedia:Connected_sum|connected sum]] of oriented manifolds. The manifold $X_{j', k_1', \dots k_n'}$ is diffeomorphic to $X_{j, k_1, \dots, k_n}$ if and only if $(j', k'_1, \dots, k'_n) = (j, k_1, \dots, k_n)$. | ||
− | An alternative complete enumeration is obtained by writing $\mathcal{M}_5(e)$ as a disjoint union | + | An alternative complete enumeration is obtained by writing $\mathcal{M}_5(e)$ as a disjoint union |
$$ \mathcal{M}_5(e) = \mathcal{M}_5^{\Spin}(e) \sqcup \mathcal{M}_5^{w_2, = \partial}(e) \sqcup \mathcal{M}_5^{w_2,\neq \partial}(e)$$ where the last two sets denote the diffeomorphism classes of non-Spinable 5-manifolds which are respectively boundaries and not boundaries. Then | $$ \mathcal{M}_5(e) = \mathcal{M}_5^{\Spin}(e) \sqcup \mathcal{M}_5^{w_2, = \partial}(e) \sqcup \mathcal{M}_5^{w_2,\neq \partial}(e)$$ where the last two sets denote the diffeomorphism classes of non-Spinable 5-manifolds which are respectively boundaries and not boundaries. Then | ||
$$ \mathcal{M}_5^{\Spin}(e) = \{ [M_G] \}, ~~\mathcal{M}_5^{w_2, = \partial}(e) = \{ [M_{(G, \omega)}] \} ~~\text{and}~~ \mathcal{M}_5^{w_2, \neq \partial}(e) = \{ [ X_{-1} \sharp M_G] \}.$$ | $$ \mathcal{M}_5^{\Spin}(e) = \{ [M_G] \}, ~~\mathcal{M}_5^{w_2, = \partial}(e) = \{ [M_{(G, \omega)}] \} ~~\text{and}~~ \mathcal{M}_5^{w_2, \neq \partial}(e) = \{ [ X_{-1} \sharp M_G] \}.$$ | ||
Line 82: | Line 86: | ||
<wikitex>; | <wikitex>; | ||
* As the invariants which classify closed, oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism. | * As the invariants which classify closed, oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism. | ||
− | + | <!-- | |
− | * The classification of simply-connected Spin 5-manifolds was one of the first applications of the [[wikipedia:H-cobordism|h-cobordism theorem]]. | + | * The classification of simply-connected Spin 5-manifolds was one of the first applications of the [[wikipedia:H-cobordism|h-cobordism theorem]]. |
− | + | --> | |
* By the construction [[#Examples_and_contructions|above]] every simply-connected, closed, smooth, Spin 5-manifold embedds into $\Rr^6$. | * By the construction [[#Examples_and_contructions|above]] every simply-connected, closed, smooth, Spin 5-manifold embedds into $\Rr^6$. | ||
Line 94: | Line 98: | ||
As $\mathcal{M}_5^{\Spin}(e) = \{[M_G]\}$ and $M_G = \partial N_G$ we see that every closed, Spin 5-manifold bounds a Spin 6-manifold. Hence the [[wikipedia:Cobordism#Cobordism_classes|bordism group]] $\Omega_5^{\Spin}$ vanishes. | As $\mathcal{M}_5^{\Spin}(e) = \{[M_G]\}$ and $M_G = \partial N_G$ we see that every closed, Spin 5-manifold bounds a Spin 6-manifold. Hence the [[wikipedia:Cobordism#Cobordism_classes|bordism group]] $\Omega_5^{\Spin}$ vanishes. | ||
− | The bordism group $\Omega_5^{\SO}$ is isomorphic to $\Zz_2$ | + | The bordism group $\Omega_5^{\SO}$ is isomorphic to $\Zz_2$, see for example {{cite|Milnor&Stasheff1974|p 203}}. Moreover this bordism group is detected by the [[wikipedia:Stiefel–Whitney_class#Stiefel.E2.80.93Whitney_numbers|Stiefel-Whitney number]] $\langle w_2(M)w_3(M), [M] \rangle \in \Zz_2$. The Wu-manifold has cohomology groups |
$$H^*(X_{-1}; \Zz_2) = \Zz_2, 0, \Zz_2, \Zz_2, 0, \Zz_2$$ and $w_2(X_{-1}) \neq 0$. It follows that $w_3(X_{-1}) = \Sq^1(w_2(X_{-1})) \neq 0$ and that $\langle w_2(X_{-1})w_3(X_{-1}), [X_{-1}] \rangle \neq 0$. We see that $[X_{-1}]$ is the generator of $\Omega_5^{\SO}$ and that a closed, smooth 5-manifold $M$ is not a boundary if and only if it is diffeomorphic to $X_{-1} \sharp M_0$ where $M_0$ is a Spin manifold. | $$H^*(X_{-1}; \Zz_2) = \Zz_2, 0, \Zz_2, \Zz_2, 0, \Zz_2$$ and $w_2(X_{-1}) \neq 0$. It follows that $w_3(X_{-1}) = \Sq^1(w_2(X_{-1})) \neq 0$ and that $\langle w_2(X_{-1})w_3(X_{-1}), [X_{-1}] \rangle \neq 0$. We see that $[X_{-1}]$ is the generator of $\Omega_5^{\SO}$ and that a closed, smooth 5-manifold $M$ is not a boundary if and only if it is diffeomorphic to $X_{-1} \sharp M_0$ where $M_0$ is a Spin manifold. | ||
− | |||
− | |||
</wikitex> | </wikitex> | ||
=== Curvature and contact structures === | === Curvature and contact structures === | ||
<wikitex>; | <wikitex>; | ||
− | * Every manifold $\sharp_r(S^2 \times S^3)$ admits a metric of positive Ricci curvature {{cite|Boyer&Galicki2006}}. | + | * Every manifold $\sharp_r(S^2 \times S^3)$ admits a metric of positive [[wikipedia:Ricci_curvature|Ricci curvature]] by {{cite|Boyer&Galicki2006}}. |
− | * Every Spin 5-manifold with the order of $TH_2(M)$ prime to 3 admits a [[wikipedia:Contact_manifold|contact structure]] {{cite|Thomas1986}}. | + | * Every Spin 5-manifold with the order of $TH_2(M)$ prime to 3 admits a [[wikipedia:Contact_manifold|contact structure]] by {{cite|Thomas1986}}. |
</wikitex> | </wikitex> | ||
Line 110: | Line 112: | ||
<wikitex>; | <wikitex>; | ||
Let $\pi_0\Diff_{+}(M)$ denote the group of [[isotopy]] classes of orientation preserving diffeomorphisms $f\co M \cong M$ and let $\Aut(H_2(M))$ be the group of isomorphisms of $H_2(M)$ preserving the linking form and the second Stiefel-Whitney class. Applying the second statement of Barden's classification [[#Classification|above]] | Let $\pi_0\Diff_{+}(M)$ denote the group of [[isotopy]] classes of orientation preserving diffeomorphisms $f\co M \cong M$ and let $\Aut(H_2(M))$ be the group of isomorphisms of $H_2(M)$ preserving the linking form and the second Stiefel-Whitney class. Applying the second statement of Barden's classification [[#Classification|above]] | ||
− | we obtain an exact sequence | + | we obtain an exact sequence |
− | $$0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\Diff_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0$$ | + | $$0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\Diff_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0$$ |
− | where $\pi_0\SDiff(M)$ is the group of isotopy classes of diffeomorphisms inducing the identity on $H_*(M)$. | + | where $\pi_0\SDiff(M)$ is the group of isotopy classes of diffeomorphisms inducing the identity on $H_*(M)$. |
− | + | <!--* Open problem: as of writing there is no computation of $\pi_0\SDiff(M)$ for a general simply-connected 5-manifold. --> | |
− | * Open problem: as of writing there is no computation of $\pi_0\SDiff(M)$ for a general simply-connected 5-manifold. | + | |
− | * | + | * There is an isomphorism $\pi_0\Diff_{+}(S^5) \cong 0$. By {{cite|Cerf1970}} and {{cite|Smale1962A}}, $\pi_0\Diff_{+}(S^5) \cong \Theta_6$, the group of [[wikipedia:Homotopy_sphere|homotopy $6$-spheres]]. But by {{cite|Keverair&Milnor1963}}, $\Theta_6 \cong 0$. |
* In the homotopy category, $\mathcal{E}_{+}(M)$, the group of homotopy classes of orientation preserving homotopy equivalences of $M$, has been extensively investigated by {{cite|Baues&Buth1996}} and is already seen to be relatively complex. | * In the homotopy category, $\mathcal{E}_{+}(M)$, the group of homotopy classes of orientation preserving homotopy equivalences of $M$, has been extensively investigated by {{cite|Baues&Buth1996}} and is already seen to be relatively complex. | ||
Line 125: | Line 126: | ||
* {{bibitem|Barden1965}} | * {{bibitem|Barden1965}} | ||
* {{bibitem|Boyer&Galicki2006}} | * {{bibitem|Boyer&Galicki2006}} | ||
+ | * {{bibitem|Cerf1970}} | ||
+ | * {{bibitem|Kervaire&Milnor1963}} | ||
* {{bibitem|Smale1962}} | * {{bibitem|Smale1962}} | ||
+ | * {{bibitem|Smale1962A}} | ||
+ | * {{bibitem|Stöcker1982}} | ||
* {{bibitem|Thomas1986}} | * {{bibitem|Thomas1986}} | ||
* {{bibitem|Wall1962}} | * {{bibitem|Wall1962}} | ||
* {{bibitem|Wall1963}} | * {{bibitem|Wall1963}} | ||
+ | * {{bibitem|Wall1964}} | ||
[[Category:Manifolds]] | [[Category:Manifolds]] | ||
[[Category:Orientable]] | [[Category:Orientable]] | ||
{{MediaWiki:Stub}} | {{MediaWiki:Stub}} |
Revision as of 14:35, 14 September 2009
An earlier version of this page was published in the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 09:22, 1 April 2011 and the changes since publication. |
Contents |
1 Introduction
Tex syntax errorand let be the subset of diffeomorphism classes of spinable manifolds. The calculation of was first obtained by Smale in [Smale1962] and was one of the first applications of the h-cobordism theorem. A little latter Barden, [Barden1965], applied a clever surgery argument and results of [Wall1964] on the diffeomorphism groups of -manifolds to give an explicit and complete classification of all of .
Simply-connected -manifolds are an appealing class of manifolds: the dimension is just large enough so that the full power of surgery techniques can be applied but it is low enough that the manifolds are simple enough to be readily classified. A feature of simply-connected -manifolds is that the homotopy, homeomorphism and diffeomorphism classification all coincide. Note that every simply-connected -dimensional Poincaré space is smoothable. The classification of of simply-connected Poincaré spaces may be found in [Stöcker1982].
2 Constructions and examples
We first list some familiar 5-manifolds using Barden's notation:
- .
- .
- , the total space of the non-trivial -bundle over .
- , the Wu-manifold, is the homogeneous space obtained from the standadard inclusion of .
- Next we present a construction of Spin 5-manifolds. Note that all homology groups are with integer coefficients. Given a finitely generated abelian group , let denote the degree 2 Moore space with . The space may be realised as a finite CW-complex and so there is an embedding . Let be a regular neighbourhood of and let be the boundary of . Then is a closed, smooth, simply-connected, spinable 5-manifold with where is the torsion subgroup of . For example, where denotes the -fold connected sum.
- For the non-Spin case let be a pair with a surjective homomorphism and as above. We shall construct a non-Spin 5-manifold with and second Stiefel-Whitney class given by composed with the projection . If let be the non-trivial -bundle over with boundary . If let be the boundary connected sum with boundary . In the general case, present where is an injective homomorphism between free abelian groups. Lift to and observe that there is a canonical identification . If is a basis for note that each is represented by a an embedded 2-sphere with trivial normal bundle. Let be the manifold obtained by attaching 3-handles to along spheres representing and let . One may check that is a non-Spin manifold as described above.
- In [Barden1965, Section 1] a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles where is a certain simply connected -manifold with boundary a simply-connected -manifold and is a diffeomorphism. Barden used results of [Wall1964] to show that diffeomorphisms realising the required ismorphisms of exist.
3 Invariants
Tex syntax error.
- be the second integral homology group of
Tex syntax error
, with torsion subgroup . - , the homomorphism defined by evaluation with the second Stiefel-Whitney class of
Tex syntax error
, . - , the smallest extended natural number such that and . If
Tex syntax error
is Spin we set .
- , the linking form of
Tex syntax error
which is a non-singular anti-symmetric bi-linear pairing on .
By [Wall1962, Proposition 1 & 2] the linking form satisfies the identity where we regard as an element of .
For example, the Wu-manifold has , non-trivial and .
An abstract non-singular, anti-symmetric linking form on a finite group is a bi-linear function such that and for all if and only if . By [Wall1963] such linking forms are classified up to isomorphism by the homomorphism . Moreover must be isomorphic to or for some finite group with if generates the summand. In particular the second Stiefel-Whitney class of a 5-manifoldTex syntax errordetermines the isomorphism class of the linking form and we see that the torsion subgroup of is of the form if or if in which case the summand is an orthogonal summand of .
4 Classification
We first present the most economical classifications of and . Let be the set of isomorphism classes finitely generated abelian groups with torsion subgroup where is trivial or and write and for the obvious subsets.
Theorem 4.2 [Barden1965]. The mapping
is an injection onto the subset of pairs where if and only if .
The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms.
Theorem 4.3 [Barden1965, Theorem 2.2]. Let and be simply-connected, closed, smooth 5-manifolds and let be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then is realised by a diffeomorphism.
This theorem can re-phrased in categorical language as follows: let be a small category, in fact groupoid, with objects where is a finitely generated abelian group, is a anti-symmetric non-singular linking form and is a homomorphism such that for all . The morphisms of are isomorphisms of abelian groups commuting with both and .
Now let be a small category, actually groupoid, with objects simply-connected, closed, smooth 5-manifolds and morphisms isotopy diffeomorphisms. Let be the funtor which mapsTex syntax errorto and to the induced map on .
Theorem 4.4 [Barden1965]. The functor
is a detecting functor. That is, it is surjective on objects and if and only if .
4.1 Enumeration
We give Barden's enumeration of the set , [Barden1965, Theorem 2.3].
- , , , .
- For , is the Spin manifold with constructed above.
- For let constructed above be the non-Spin manifold with .
With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by
where , , divides or and denotes the connected sum of oriented manifolds. The manifold is diffeomorphic to if and only if .
An alternative complete enumeration is obtained by writing as a disjoint union
5 Further discussion
- As the invariants which classify closed, oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism.
- By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into .
- As the invariants for are isomorphic to the invariants of
Tex syntax error
we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.
5.1 Bordism groups
As and we see that every closed, Spin 5-manifold bounds a Spin 6-manifold. Hence the bordism group vanishes.
The bordism group is isomorphic to , see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number . The Wu-manifold has cohomology groups
Tex syntax erroris not a boundary if and only if it is diffeomorphic to where is a Spin manifold.
5.2 Curvature and contact structures
- Every manifold admits a metric of positive Ricci curvature by [Boyer&Galicki2006].
- Every Spin 5-manifold with the order of prime to 3 admits a contact structure by [Thomas1986].
5.3 Mapping class groups
Let denote the group of isotopy classes of orientation preserving diffeomorphisms and let be the group of isomorphisms of preserving the linking form and the second Stiefel-Whitney class. Applying the second statement of Barden's classification above we obtain an exact sequence
where is the group of isotopy classes of diffeomorphisms inducing the identity on .
- There is an isomphorism . By [Cerf1970] and [Smale1962A], , the group of homotopy -spheres. But by [Keverair&Milnor1963], .
- In the homotopy category, , the group of homotopy classes of orientation preserving homotopy equivalences of
Tex syntax error
, has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
6 References
- [Baues&Buth1996] H. J. Baues and J. Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996), no.4, 573–614. MR1406269 (97g:55009) Zbl 0881.55008
- [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
- [Boyer&Galicki2006] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235 (electronic). MR2284055 (2007k:53057) Zbl 1129.53026
- [Cerf1970] J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. (1970), no.39, 5–173. MR0292089 (45 #1176) Zbl 0213.25202
- [Kervaire&Milnor1963] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR0148075 (26 #5584) Zbl 0115.40505
- [Smale1962] S. Smale, On the structure of -manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
- [Smale1962A] Template:Smale1962A
- [Stöcker1982] R. Stöcker, On the structure of -dimensional Poincaré duality spaces, Comment. Math. Helv. 57 (1982), no.3, 481–510. MR689075 (85b:57022) Zbl 0507.57012
- [Thomas1986] C. B. Thomas, Contact structures on -connected -manifolds, 18 (1986), 255–270. MR925869 (89b:53074) Zbl 0642.57014
- [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803
- [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903
- [Wall1964] C. T. C. Wall, Diffeomorphisms of -manifolds, J. London Math. Soc. 39 (1964), 131–140. MR0163323 (29 #626) Zbl 0121.18101MediaWiki:Stub
Tex syntax errorand let be the subset of diffeomorphism classes of spinable manifolds. The calculation of was first obtained by Smale in [Smale1962] and was one of the first applications of the h-cobordism theorem. A little latter Barden, [Barden1965], applied a clever surgery argument and results of [Wall1964] on the diffeomorphism groups of -manifolds to give an explicit and complete classification of all of .
Simply-connected -manifolds are an appealing class of manifolds: the dimension is just large enough so that the full power of surgery techniques can be applied but it is low enough that the manifolds are simple enough to be readily classified. A feature of simply-connected -manifolds is that the homotopy, homeomorphism and diffeomorphism classification all coincide. Note that every simply-connected -dimensional Poincaré space is smoothable. The classification of of simply-connected Poincaré spaces may be found in [Stöcker1982].
2 Constructions and examples
We first list some familiar 5-manifolds using Barden's notation:
- .
- .
- , the total space of the non-trivial -bundle over .
- , the Wu-manifold, is the homogeneous space obtained from the standadard inclusion of .
- Next we present a construction of Spin 5-manifolds. Note that all homology groups are with integer coefficients. Given a finitely generated abelian group , let denote the degree 2 Moore space with . The space may be realised as a finite CW-complex and so there is an embedding . Let be a regular neighbourhood of and let be the boundary of . Then is a closed, smooth, simply-connected, spinable 5-manifold with where is the torsion subgroup of . For example, where denotes the -fold connected sum.
- For the non-Spin case let be a pair with a surjective homomorphism and as above. We shall construct a non-Spin 5-manifold with and second Stiefel-Whitney class given by composed with the projection . If let be the non-trivial -bundle over with boundary . If let be the boundary connected sum with boundary . In the general case, present where is an injective homomorphism between free abelian groups. Lift to and observe that there is a canonical identification . If is a basis for note that each is represented by a an embedded 2-sphere with trivial normal bundle. Let be the manifold obtained by attaching 3-handles to along spheres representing and let . One may check that is a non-Spin manifold as described above.
- In [Barden1965, Section 1] a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles where is a certain simply connected -manifold with boundary a simply-connected -manifold and is a diffeomorphism. Barden used results of [Wall1964] to show that diffeomorphisms realising the required ismorphisms of exist.
3 Invariants
Tex syntax error.
- be the second integral homology group of
Tex syntax error
, with torsion subgroup . - , the homomorphism defined by evaluation with the second Stiefel-Whitney class of
Tex syntax error
, . - , the smallest extended natural number such that and . If
Tex syntax error
is Spin we set .
- , the linking form of
Tex syntax error
which is a non-singular anti-symmetric bi-linear pairing on .
By [Wall1962, Proposition 1 & 2] the linking form satisfies the identity where we regard as an element of .
For example, the Wu-manifold has , non-trivial and .
An abstract non-singular, anti-symmetric linking form on a finite group is a bi-linear function such that and for all if and only if . By [Wall1963] such linking forms are classified up to isomorphism by the homomorphism . Moreover must be isomorphic to or for some finite group with if generates the summand. In particular the second Stiefel-Whitney class of a 5-manifoldTex syntax errordetermines the isomorphism class of the linking form and we see that the torsion subgroup of is of the form if or if in which case the summand is an orthogonal summand of .
4 Classification
We first present the most economical classifications of and . Let be the set of isomorphism classes finitely generated abelian groups with torsion subgroup where is trivial or and write and for the obvious subsets.
Theorem 4.2 [Barden1965]. The mapping
is an injection onto the subset of pairs where if and only if .
The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms.
Theorem 4.3 [Barden1965, Theorem 2.2]. Let and be simply-connected, closed, smooth 5-manifolds and let be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then is realised by a diffeomorphism.
This theorem can re-phrased in categorical language as follows: let be a small category, in fact groupoid, with objects where is a finitely generated abelian group, is a anti-symmetric non-singular linking form and is a homomorphism such that for all . The morphisms of are isomorphisms of abelian groups commuting with both and .
Now let be a small category, actually groupoid, with objects simply-connected, closed, smooth 5-manifolds and morphisms isotopy diffeomorphisms. Let be the funtor which mapsTex syntax errorto and to the induced map on .
Theorem 4.4 [Barden1965]. The functor
is a detecting functor. That is, it is surjective on objects and if and only if .
4.1 Enumeration
We give Barden's enumeration of the set , [Barden1965, Theorem 2.3].
- , , , .
- For , is the Spin manifold with constructed above.
- For let constructed above be the non-Spin manifold with .
With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by
where , , divides or and denotes the connected sum of oriented manifolds. The manifold is diffeomorphic to if and only if .
An alternative complete enumeration is obtained by writing as a disjoint union
5 Further discussion
- As the invariants which classify closed, oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism.
- By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into .
- As the invariants for are isomorphic to the invariants of
Tex syntax error
we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.
5.1 Bordism groups
As and we see that every closed, Spin 5-manifold bounds a Spin 6-manifold. Hence the bordism group vanishes.
The bordism group is isomorphic to , see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number . The Wu-manifold has cohomology groups
Tex syntax erroris not a boundary if and only if it is diffeomorphic to where is a Spin manifold.
5.2 Curvature and contact structures
- Every manifold admits a metric of positive Ricci curvature by [Boyer&Galicki2006].
- Every Spin 5-manifold with the order of prime to 3 admits a contact structure by [Thomas1986].
5.3 Mapping class groups
Let denote the group of isotopy classes of orientation preserving diffeomorphisms and let be the group of isomorphisms of preserving the linking form and the second Stiefel-Whitney class. Applying the second statement of Barden's classification above we obtain an exact sequence
where is the group of isotopy classes of diffeomorphisms inducing the identity on .
- There is an isomphorism . By [Cerf1970] and [Smale1962A], , the group of homotopy -spheres. But by [Keverair&Milnor1963], .
- In the homotopy category, , the group of homotopy classes of orientation preserving homotopy equivalences of
Tex syntax error
, has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
6 References
- [Baues&Buth1996] H. J. Baues and J. Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996), no.4, 573–614. MR1406269 (97g:55009) Zbl 0881.55008
- [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
- [Boyer&Galicki2006] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235 (electronic). MR2284055 (2007k:53057) Zbl 1129.53026
- [Cerf1970] J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. (1970), no.39, 5–173. MR0292089 (45 #1176) Zbl 0213.25202
- [Kervaire&Milnor1963] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR0148075 (26 #5584) Zbl 0115.40505
- [Smale1962] S. Smale, On the structure of -manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
- [Smale1962A] Template:Smale1962A
- [Stöcker1982] R. Stöcker, On the structure of -dimensional Poincaré duality spaces, Comment. Math. Helv. 57 (1982), no.3, 481–510. MR689075 (85b:57022) Zbl 0507.57012
- [Thomas1986] C. B. Thomas, Contact structures on -connected -manifolds, 18 (1986), 255–270. MR925869 (89b:53074) Zbl 0642.57014
- [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803
- [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903
- [Wall1964] C. T. C. Wall, Diffeomorphisms of -manifolds, J. London Math. Soc. 39 (1964), 131–140. MR0163323 (29 #626) Zbl 0121.18101MediaWiki:Stub
Tex syntax errorand let be the subset of diffeomorphism classes of spinable manifolds. The calculation of was first obtained by Smale in [Smale1962] and was one of the first applications of the h-cobordism theorem. A little latter Barden, [Barden1965], applied a clever surgery argument and results of [Wall1964] on the diffeomorphism groups of -manifolds to give an explicit and complete classification of all of .
Simply-connected -manifolds are an appealing class of manifolds: the dimension is just large enough so that the full power of surgery techniques can be applied but it is low enough that the manifolds are simple enough to be readily classified. A feature of simply-connected -manifolds is that the homotopy, homeomorphism and diffeomorphism classification all coincide. Note that every simply-connected -dimensional Poincaré space is smoothable. The classification of of simply-connected Poincaré spaces may be found in [Stöcker1982].
2 Constructions and examples
We first list some familiar 5-manifolds using Barden's notation:
- .
- .
- , the total space of the non-trivial -bundle over .
- , the Wu-manifold, is the homogeneous space obtained from the standadard inclusion of .
- Next we present a construction of Spin 5-manifolds. Note that all homology groups are with integer coefficients. Given a finitely generated abelian group , let denote the degree 2 Moore space with . The space may be realised as a finite CW-complex and so there is an embedding . Let be a regular neighbourhood of and let be the boundary of . Then is a closed, smooth, simply-connected, spinable 5-manifold with where is the torsion subgroup of . For example, where denotes the -fold connected sum.
- For the non-Spin case let be a pair with a surjective homomorphism and as above. We shall construct a non-Spin 5-manifold with and second Stiefel-Whitney class given by composed with the projection . If let be the non-trivial -bundle over with boundary . If let be the boundary connected sum with boundary . In the general case, present where is an injective homomorphism between free abelian groups. Lift to and observe that there is a canonical identification . If is a basis for note that each is represented by a an embedded 2-sphere with trivial normal bundle. Let be the manifold obtained by attaching 3-handles to along spheres representing and let . One may check that is a non-Spin manifold as described above.
- In [Barden1965, Section 1] a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles where is a certain simply connected -manifold with boundary a simply-connected -manifold and is a diffeomorphism. Barden used results of [Wall1964] to show that diffeomorphisms realising the required ismorphisms of exist.
3 Invariants
Tex syntax error.
- be the second integral homology group of
Tex syntax error
, with torsion subgroup . - , the homomorphism defined by evaluation with the second Stiefel-Whitney class of
Tex syntax error
, . - , the smallest extended natural number such that and . If
Tex syntax error
is Spin we set .
- , the linking form of
Tex syntax error
which is a non-singular anti-symmetric bi-linear pairing on .
By [Wall1962, Proposition 1 & 2] the linking form satisfies the identity where we regard as an element of .
For example, the Wu-manifold has , non-trivial and .
An abstract non-singular, anti-symmetric linking form on a finite group is a bi-linear function such that and for all if and only if . By [Wall1963] such linking forms are classified up to isomorphism by the homomorphism . Moreover must be isomorphic to or for some finite group with if generates the summand. In particular the second Stiefel-Whitney class of a 5-manifoldTex syntax errordetermines the isomorphism class of the linking form and we see that the torsion subgroup of is of the form if or if in which case the summand is an orthogonal summand of .
4 Classification
We first present the most economical classifications of and . Let be the set of isomorphism classes finitely generated abelian groups with torsion subgroup where is trivial or and write and for the obvious subsets.
Theorem 4.2 [Barden1965]. The mapping
is an injection onto the subset of pairs where if and only if .
The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms.
Theorem 4.3 [Barden1965, Theorem 2.2]. Let and be simply-connected, closed, smooth 5-manifolds and let be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then is realised by a diffeomorphism.
This theorem can re-phrased in categorical language as follows: let be a small category, in fact groupoid, with objects where is a finitely generated abelian group, is a anti-symmetric non-singular linking form and is a homomorphism such that for all . The morphisms of are isomorphisms of abelian groups commuting with both and .
Now let be a small category, actually groupoid, with objects simply-connected, closed, smooth 5-manifolds and morphisms isotopy diffeomorphisms. Let be the funtor which mapsTex syntax errorto and to the induced map on .
Theorem 4.4 [Barden1965]. The functor
is a detecting functor. That is, it is surjective on objects and if and only if .
4.1 Enumeration
We give Barden's enumeration of the set , [Barden1965, Theorem 2.3].
- , , , .
- For , is the Spin manifold with constructed above.
- For let constructed above be the non-Spin manifold with .
With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by
where , , divides or and denotes the connected sum of oriented manifolds. The manifold is diffeomorphic to if and only if .
An alternative complete enumeration is obtained by writing as a disjoint union
5 Further discussion
- As the invariants which classify closed, oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism.
- By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into .
- As the invariants for are isomorphic to the invariants of
Tex syntax error
we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.
5.1 Bordism groups
As and we see that every closed, Spin 5-manifold bounds a Spin 6-manifold. Hence the bordism group vanishes.
The bordism group is isomorphic to , see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number . The Wu-manifold has cohomology groups
Tex syntax erroris not a boundary if and only if it is diffeomorphic to where is a Spin manifold.
5.2 Curvature and contact structures
- Every manifold admits a metric of positive Ricci curvature by [Boyer&Galicki2006].
- Every Spin 5-manifold with the order of prime to 3 admits a contact structure by [Thomas1986].
5.3 Mapping class groups
Let denote the group of isotopy classes of orientation preserving diffeomorphisms and let be the group of isomorphisms of preserving the linking form and the second Stiefel-Whitney class. Applying the second statement of Barden's classification above we obtain an exact sequence
where is the group of isotopy classes of diffeomorphisms inducing the identity on .
- There is an isomphorism . By [Cerf1970] and [Smale1962A], , the group of homotopy -spheres. But by [Keverair&Milnor1963], .
- In the homotopy category, , the group of homotopy classes of orientation preserving homotopy equivalences of
Tex syntax error
, has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
6 References
- [Baues&Buth1996] H. J. Baues and J. Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996), no.4, 573–614. MR1406269 (97g:55009) Zbl 0881.55008
- [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
- [Boyer&Galicki2006] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235 (electronic). MR2284055 (2007k:53057) Zbl 1129.53026
- [Cerf1970] J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. (1970), no.39, 5–173. MR0292089 (45 #1176) Zbl 0213.25202
- [Kervaire&Milnor1963] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR0148075 (26 #5584) Zbl 0115.40505
- [Smale1962] S. Smale, On the structure of -manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
- [Smale1962A] Template:Smale1962A
- [Stöcker1982] R. Stöcker, On the structure of -dimensional Poincaré duality spaces, Comment. Math. Helv. 57 (1982), no.3, 481–510. MR689075 (85b:57022) Zbl 0507.57012
- [Thomas1986] C. B. Thomas, Contact structures on -connected -manifolds, 18 (1986), 255–270. MR925869 (89b:53074) Zbl 0642.57014
- [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803
- [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903
- [Wall1964] C. T. C. Wall, Diffeomorphisms of -manifolds, J. London Math. Soc. 39 (1964), 131–140. MR0163323 (29 #626) Zbl 0121.18101MediaWiki:Stub
Tex syntax errorand let be the subset of diffeomorphism classes of spinable manifolds. The calculation of was first obtained by Smale in [Smale1962] and was one of the first applications of the h-cobordism theorem. A little latter Barden, [Barden1965], applied a clever surgery argument and results of [Wall1964] on the diffeomorphism groups of -manifolds to give an explicit and complete classification of all of .
Simply-connected -manifolds are an appealing class of manifolds: the dimension is just large enough so that the full power of surgery techniques can be applied but it is low enough that the manifolds are simple enough to be readily classified. A feature of simply-connected -manifolds is that the homotopy, homeomorphism and diffeomorphism classification all coincide. Note that every simply-connected -dimensional Poincaré space is smoothable. The classification of of simply-connected Poincaré spaces may be found in [Stöcker1982].
2 Constructions and examples
We first list some familiar 5-manifolds using Barden's notation:
- .
- .
- , the total space of the non-trivial -bundle over .
- , the Wu-manifold, is the homogeneous space obtained from the standadard inclusion of .
- Next we present a construction of Spin 5-manifolds. Note that all homology groups are with integer coefficients. Given a finitely generated abelian group , let denote the degree 2 Moore space with . The space may be realised as a finite CW-complex and so there is an embedding . Let be a regular neighbourhood of and let be the boundary of . Then is a closed, smooth, simply-connected, spinable 5-manifold with where is the torsion subgroup of . For example, where denotes the -fold connected sum.
- For the non-Spin case let be a pair with a surjective homomorphism and as above. We shall construct a non-Spin 5-manifold with and second Stiefel-Whitney class given by composed with the projection . If let be the non-trivial -bundle over with boundary . If let be the boundary connected sum with boundary . In the general case, present where is an injective homomorphism between free abelian groups. Lift to and observe that there is a canonical identification . If is a basis for note that each is represented by a an embedded 2-sphere with trivial normal bundle. Let be the manifold obtained by attaching 3-handles to along spheres representing and let . One may check that is a non-Spin manifold as described above.
- In [Barden1965, Section 1] a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles where is a certain simply connected -manifold with boundary a simply-connected -manifold and is a diffeomorphism. Barden used results of [Wall1964] to show that diffeomorphisms realising the required ismorphisms of exist.
3 Invariants
Tex syntax error.
- be the second integral homology group of
Tex syntax error
, with torsion subgroup . - , the homomorphism defined by evaluation with the second Stiefel-Whitney class of
Tex syntax error
, . - , the smallest extended natural number such that and . If
Tex syntax error
is Spin we set .
- , the linking form of
Tex syntax error
which is a non-singular anti-symmetric bi-linear pairing on .
By [Wall1962, Proposition 1 & 2] the linking form satisfies the identity where we regard as an element of .
For example, the Wu-manifold has , non-trivial and .
An abstract non-singular, anti-symmetric linking form on a finite group is a bi-linear function such that and for all if and only if . By [Wall1963] such linking forms are classified up to isomorphism by the homomorphism . Moreover must be isomorphic to or for some finite group with if generates the summand. In particular the second Stiefel-Whitney class of a 5-manifoldTex syntax errordetermines the isomorphism class of the linking form and we see that the torsion subgroup of is of the form if or if in which case the summand is an orthogonal summand of .
4 Classification
We first present the most economical classifications of and . Let be the set of isomorphism classes finitely generated abelian groups with torsion subgroup where is trivial or and write and for the obvious subsets.
Theorem 4.2 [Barden1965]. The mapping
is an injection onto the subset of pairs where if and only if .
The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms.
Theorem 4.3 [Barden1965, Theorem 2.2]. Let and be simply-connected, closed, smooth 5-manifolds and let be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then is realised by a diffeomorphism.
This theorem can re-phrased in categorical language as follows: let be a small category, in fact groupoid, with objects where is a finitely generated abelian group, is a anti-symmetric non-singular linking form and is a homomorphism such that for all . The morphisms of are isomorphisms of abelian groups commuting with both and .
Now let be a small category, actually groupoid, with objects simply-connected, closed, smooth 5-manifolds and morphisms isotopy diffeomorphisms. Let be the funtor which mapsTex syntax errorto and to the induced map on .
Theorem 4.4 [Barden1965]. The functor
is a detecting functor. That is, it is surjective on objects and if and only if .
4.1 Enumeration
We give Barden's enumeration of the set , [Barden1965, Theorem 2.3].
- , , , .
- For , is the Spin manifold with constructed above.
- For let constructed above be the non-Spin manifold with .
With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by
where , , divides or and denotes the connected sum of oriented manifolds. The manifold is diffeomorphic to if and only if .
An alternative complete enumeration is obtained by writing as a disjoint union
5 Further discussion
- As the invariants which classify closed, oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism.
- By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into .
- As the invariants for are isomorphic to the invariants of
Tex syntax error
we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.
5.1 Bordism groups
As and we see that every closed, Spin 5-manifold bounds a Spin 6-manifold. Hence the bordism group vanishes.
The bordism group is isomorphic to , see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number . The Wu-manifold has cohomology groups
Tex syntax erroris not a boundary if and only if it is diffeomorphic to where is a Spin manifold.
5.2 Curvature and contact structures
- Every manifold admits a metric of positive Ricci curvature by [Boyer&Galicki2006].
- Every Spin 5-manifold with the order of prime to 3 admits a contact structure by [Thomas1986].
5.3 Mapping class groups
Let denote the group of isotopy classes of orientation preserving diffeomorphisms and let be the group of isomorphisms of preserving the linking form and the second Stiefel-Whitney class. Applying the second statement of Barden's classification above we obtain an exact sequence
where is the group of isotopy classes of diffeomorphisms inducing the identity on .
- There is an isomphorism . By [Cerf1970] and [Smale1962A], , the group of homotopy -spheres. But by [Keverair&Milnor1963], .
- In the homotopy category, , the group of homotopy classes of orientation preserving homotopy equivalences of
Tex syntax error
, has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
6 References
- [Baues&Buth1996] H. J. Baues and J. Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996), no.4, 573–614. MR1406269 (97g:55009) Zbl 0881.55008
- [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
- [Boyer&Galicki2006] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235 (electronic). MR2284055 (2007k:53057) Zbl 1129.53026
- [Cerf1970] J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. (1970), no.39, 5–173. MR0292089 (45 #1176) Zbl 0213.25202
- [Kervaire&Milnor1963] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR0148075 (26 #5584) Zbl 0115.40505
- [Smale1962] S. Smale, On the structure of -manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
- [Smale1962A] Template:Smale1962A
- [Stöcker1982] R. Stöcker, On the structure of -dimensional Poincaré duality spaces, Comment. Math. Helv. 57 (1982), no.3, 481–510. MR689075 (85b:57022) Zbl 0507.57012
- [Thomas1986] C. B. Thomas, Contact structures on -connected -manifolds, 18 (1986), 255–270. MR925869 (89b:53074) Zbl 0642.57014
- [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803
- [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903
- [Wall1964] C. T. C. Wall, Diffeomorphisms of -manifolds, J. London Math. Soc. 39 (1964), 131–140. MR0163323 (29 #626) Zbl 0121.18101MediaWiki:Stub
Tex syntax errorand let be the subset of diffeomorphism classes of spinable manifolds. The calculation of was first obtained by Smale in [Smale1962] and was one of the first applications of the h-cobordism theorem. A little latter Barden, [Barden1965], applied a clever surgery argument and results of [Wall1964] on the diffeomorphism groups of -manifolds to give an explicit and complete classification of all of .
Simply-connected -manifolds are an appealing class of manifolds: the dimension is just large enough so that the full power of surgery techniques can be applied but it is low enough that the manifolds are simple enough to be readily classified. A feature of simply-connected -manifolds is that the homotopy, homeomorphism and diffeomorphism classification all coincide. Note that every simply-connected -dimensional Poincaré space is smoothable. The classification of of simply-connected Poincaré spaces may be found in [Stöcker1982].
2 Constructions and examples
We first list some familiar 5-manifolds using Barden's notation:
- .
- .
- , the total space of the non-trivial -bundle over .
- , the Wu-manifold, is the homogeneous space obtained from the standadard inclusion of .
- Next we present a construction of Spin 5-manifolds. Note that all homology groups are with integer coefficients. Given a finitely generated abelian group , let denote the degree 2 Moore space with . The space may be realised as a finite CW-complex and so there is an embedding . Let be a regular neighbourhood of and let be the boundary of . Then is a closed, smooth, simply-connected, spinable 5-manifold with where is the torsion subgroup of . For example, where denotes the -fold connected sum.
- For the non-Spin case let be a pair with a surjective homomorphism and as above. We shall construct a non-Spin 5-manifold with and second Stiefel-Whitney class given by composed with the projection . If let be the non-trivial -bundle over with boundary . If let be the boundary connected sum with boundary . In the general case, present where is an injective homomorphism between free abelian groups. Lift to and observe that there is a canonical identification . If is a basis for note that each is represented by a an embedded 2-sphere with trivial normal bundle. Let be the manifold obtained by attaching 3-handles to along spheres representing and let . One may check that is a non-Spin manifold as described above.
- In [Barden1965, Section 1] a construction of simply-connected 5-manifolds is given by expressing them as twisted doubles where is a certain simply connected -manifold with boundary a simply-connected -manifold and is a diffeomorphism. Barden used results of [Wall1964] to show that diffeomorphisms realising the required ismorphisms of exist.
3 Invariants
Tex syntax error.
- be the second integral homology group of
Tex syntax error
, with torsion subgroup . - , the homomorphism defined by evaluation with the second Stiefel-Whitney class of
Tex syntax error
, . - , the smallest extended natural number such that and . If
Tex syntax error
is Spin we set .
- , the linking form of
Tex syntax error
which is a non-singular anti-symmetric bi-linear pairing on .
By [Wall1962, Proposition 1 & 2] the linking form satisfies the identity where we regard as an element of .
For example, the Wu-manifold has , non-trivial and .
An abstract non-singular, anti-symmetric linking form on a finite group is a bi-linear function such that and for all if and only if . By [Wall1963] such linking forms are classified up to isomorphism by the homomorphism . Moreover must be isomorphic to or for some finite group with if generates the summand. In particular the second Stiefel-Whitney class of a 5-manifoldTex syntax errordetermines the isomorphism class of the linking form and we see that the torsion subgroup of is of the form if or if in which case the summand is an orthogonal summand of .
4 Classification
We first present the most economical classifications of and . Let be the set of isomorphism classes finitely generated abelian groups with torsion subgroup where is trivial or and write and for the obvious subsets.
Theorem 4.2 [Barden1965]. The mapping
is an injection onto the subset of pairs where if and only if .
The above theorems follow from the following theorem of Barden and the classification of anti-symmetric linking forms.
Theorem 4.3 [Barden1965, Theorem 2.2]. Let and be simply-connected, closed, smooth 5-manifolds and let be an isomorphism preserving the linking form and the second Stiefel-Whitney class. Then is realised by a diffeomorphism.
This theorem can re-phrased in categorical language as follows: let be a small category, in fact groupoid, with objects where is a finitely generated abelian group, is a anti-symmetric non-singular linking form and is a homomorphism such that for all . The morphisms of are isomorphisms of abelian groups commuting with both and .
Now let be a small category, actually groupoid, with objects simply-connected, closed, smooth 5-manifolds and morphisms isotopy diffeomorphisms. Let be the funtor which mapsTex syntax errorto and to the induced map on .
Theorem 4.4 [Barden1965]. The functor
is a detecting functor. That is, it is surjective on objects and if and only if .
4.1 Enumeration
We give Barden's enumeration of the set , [Barden1965, Theorem 2.3].
- , , , .
- For , is the Spin manifold with constructed above.
- For let constructed above be the non-Spin manifold with .
With this notation [Barden1965, Theorem 2.3] states that a complete list of diffeomorphism classes of simply-connected 5-manifolds is given by
where , , divides or and denotes the connected sum of oriented manifolds. The manifold is diffeomorphic to if and only if .
An alternative complete enumeration is obtained by writing as a disjoint union
5 Further discussion
- As the invariants which classify closed, oriented 5-manifolds are homotopy invariants, we see that the same classification holds up to homotopy, homeomorphism and piecewise linear homeomorphism.
- By the construction above every simply-connected, closed, smooth, Spin 5-manifold embedds into .
- As the invariants for are isomorphic to the invariants of
Tex syntax error
we see that every smooth 5-manifold admits an orientation reversing diffeomorphism: i.e. all 5-manifolds are smoothly amphicheiral.
5.1 Bordism groups
As and we see that every closed, Spin 5-manifold bounds a Spin 6-manifold. Hence the bordism group vanishes.
The bordism group is isomorphic to , see for example [Milnor&Stasheff1974, p 203]. Moreover this bordism group is detected by the Stiefel-Whitney number . The Wu-manifold has cohomology groups
Tex syntax erroris not a boundary if and only if it is diffeomorphic to where is a Spin manifold.
5.2 Curvature and contact structures
- Every manifold admits a metric of positive Ricci curvature by [Boyer&Galicki2006].
- Every Spin 5-manifold with the order of prime to 3 admits a contact structure by [Thomas1986].
5.3 Mapping class groups
Let denote the group of isotopy classes of orientation preserving diffeomorphisms and let be the group of isomorphisms of preserving the linking form and the second Stiefel-Whitney class. Applying the second statement of Barden's classification above we obtain an exact sequence
where is the group of isotopy classes of diffeomorphisms inducing the identity on .
- There is an isomphorism . By [Cerf1970] and [Smale1962A], , the group of homotopy -spheres. But by [Keverair&Milnor1963], .
- In the homotopy category, , the group of homotopy classes of orientation preserving homotopy equivalences of
Tex syntax error
, has been extensively investigated by [Baues&Buth1996] and is already seen to be relatively complex.
6 References
- [Baues&Buth1996] H. J. Baues and J. Buth, On the group of homotopy equivalences of simply connected five manifolds, Math. Z. 222 (1996), no.4, 573–614. MR1406269 (97g:55009) Zbl 0881.55008
- [Barden1965] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385. MR0184241 (32 #1714) Zbl 0136.20602
- [Boyer&Galicki2006] C. P. Boyer and K. Galicki, Highly connected manifolds with positive Ricci curvature, Geom. Topol. 10 (2006), 2219–2235 (electronic). MR2284055 (2007k:53057) Zbl 1129.53026
- [Cerf1970] J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. (1970), no.39, 5–173. MR0292089 (45 #1176) Zbl 0213.25202
- [Kervaire&Milnor1963] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR0148075 (26 #5584) Zbl 0115.40505
- [Smale1962] S. Smale, On the structure of -manifolds, Ann. of Math. (2) 75 (1962), 38–46. MR0141133 (25 #4544) Zbl 0101.16103
- [Smale1962A] Template:Smale1962A
- [Stöcker1982] R. Stöcker, On the structure of -dimensional Poincaré duality spaces, Comment. Math. Helv. 57 (1982), no.3, 481–510. MR689075 (85b:57022) Zbl 0507.57012
- [Thomas1986] C. B. Thomas, Contact structures on -connected -manifolds, 18 (1986), 255–270. MR925869 (89b:53074) Zbl 0642.57014
- [Wall1962] C. T. C. Wall, Killing the middle homotopy groups of odd dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421–433. MR0139185 (25 #2621) Zbl 0199.26803
- [Wall1963] C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298. MR0156890 (28 #133) Zbl 0215.39903
- [Wall1964] C. T. C. Wall, Diffeomorphisms of -manifolds, J. London Math. Soc. 39 (1964), 131–140. MR0163323 (29 #626) Zbl 0121.18101MediaWiki:Stub